Weak Convergence of the Empirical Spectral Distribution of High-Dimensional Band Sample Covariance Matrices

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In this article, we investigate high-dimensional band sample covariance matrices under the regime that the sample size n, the dimension p, and the bandwidth d tend simultaneously to infinity such that $$\begin{aligned} n/p\rightarrow 0 \ \ \text {and} \ \ d/n\rightarrow y>0. \end{aligned}$$ It is shown that the empirical spectral distribution of those matrices converges weakly to a deterministic probability measure with probability 1. The limiting measure is characterized by its moments. Certain restricted compositions of natural numbers play a crucial role in the evaluation of the expected moments of the empirical spectral distribution.